Quantum deformed magnon kinematics

The dispersion relation for planar N = 4 supersymmetric Yang-Mills is identified with the Casimir of a quantum deformed two-dimensional kinematical symmetry, Eq(1, 1). The quantum deformed symmetry algebra is generated by the momentum, energy and boost, with deformation parameter q = e. Representing the boost as the infinitesimal generator for translations on the rapidity space leads to an elliptic uniformization with crossing transformations implemented through translations by the elliptic half-periods. This quantum deformed algebra can be interpreted as the kinematical symmetry of a discrete integrable model with lattice spacing given by the BMN length a = 2π/ √ λ. The interpretation of the boost generator as the corner transfer matrix is briefly discussed. Introduction. An important boost into our current understanding of the AdS/CFT correspondence came from the BMN suggestion to probe sectors with large quantum numbers [1]. The BMN limit provided also an appealing dispersion relation for planar N = 4 supersymmetric Yang-Mills. The uncovering of integrability both on the gauge [2] and string sides [3] of the correspondence allowed then the search for the explicit form of the scattering matrices of N = 4 Yang-Mills [4] and of type IIB string theory in AdS5×S [5]. The construction in [4] also implied a derivation in purely algebraic terms of a general dispersion relation of BMN type. This dispersion relation exhibits some sort of double nature, as it looks relativistic in a certain limit, while also includes typical aspects of a lattice dispersion relation. The absence of conventional relativistic invariance is indeed a feature of magnon kinematics in the AdS/CFT correspondence, and requires an elliptic approach to crossing symmetry in the scattering matrix. In [6] an elliptic uniformization was derived and shown to lead to a non-trivial implementation of crossing in terms of translations by half-periods of the elliptic curve defining the kinematical rapidity plane. In this note we address the problem of the kinematical origin of the BMN type of dispersion relations by identifying the kinematical symmetry group underlying the integrable model. This symmetry is a quantum deformation of the pseudoeuclidean group Eq(1, 1) [7], with the deformation parameter q given in terms of the ‘t Hooft coupling constant by q = e. The Casimir of this algebra is indeed the dispersion relation in N = 4 supersymmetric Yang-Mills, and the boost is the generator of infinitesimal translations on the elliptic rapidity plane. The meaning of this kinematical symmetry must be understood from the structure of the Hopf algebra symmetry [8, 9, 10, 5, 11, 12]. In [8] the existence of a central Hopf subalgebra was noticed and the spectrum of this center was proposed as the rapidity plane. This is indeed the usual situation in integrable models of the chiral Potts type, where the kinematical symmetry group acts naturally on the spectrum of the central Hopf subalgebra. The kinematical symmetry Eq(1, 1) has non-trivial co-multiplications for the boost generators that are at the root of the elliptic nature of the rapidity space. The comultiplication rules also underly the non-trivial crossing transformations on the rapidities. Furthermore, as pointed out in [13, 14], these quantum deformed algebras are the natural candidates to kinematical symmetry groups of lattice models, with the lattice spacing being related to the quantum deformation parameter. As we will show in the case of N = 4 Yang-Mills this lattice spacing can be identified with the scale introduced in [1] through